A student seeks to prove that there exist infinitely many pairs of non-zero integers such
that a particular third degree polynomial is square in the ring of integers. Since the
exercise appears in a chapter on Galois theory, Doctor Jacques expands the scope of
the question to proving that there are infinitely many such polynomials.

Let G be an Abelian group. Show that the elements of finite order in G
form a subgroup. This subgroup is called the torsion subgroup of G.
Now find the torsion subgroup of the multiplicative group R* of
nonzero real numbers.

Why are polynomials whose only common factors are constants considered
'relatively prime'? Why are the common constants not considered? For
example, 3x + 6 and 3x^2 + 12 are considered relatively prime even
though they have a common constant factor of 3.